We present exact solution of an electrostatic problem in terms of the potential distribution and density profile of an incompletely depleted two-dimensional electron gas in a heterostructure covered by a half-infinite gate. Using this solution, we calculate configurations of compressible and incompressible (insulating) strips that are formed in the presence of a quantizing magnetic field. We discuss the application of our results to transport experiments on the inter-Landau-level and spin diodes, as well as edge-magnetoplasmon propagation.